An o-minimal structure which does not admit C∞ cellular decomposition
[Une structure o-minimale sans décomposition cellulaire lisse]
Le Gal, Olivier1 ; Rolin, Jean-Philippe2
1 University of Toronto Department of Mathematics Toronto, Ontario M5S 2E4 (Canada)
2 Université de Bourgogne IMB, UFR Sciences et Techniques 9, Avenue Alain Savary BP 47870 21078 Dijon (France)
Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 543-562.

Nous présentons un exemple de structure o-minimale n’admettant pas la propriété de décomposition cellulaire C∞. Pour ce faire, nous construisons une fonction H dont le germe en 0 admet un représentant Ck pour tout entier k, mais n’admet aucun représentant C∞. Une condition de transcendance sur les coefficients de la série de Taylor de H assure alors la quasi-analyticité de certaines algèbres différentielles 𝒜n(H) engendrées par H. La o-minimalité de la structure engendrée par H est enfin déduite de cette quasi-analyticité.

We present an example of an o-minimal structure which does not admit C∞ cellular decomposition. To this end, we construct a function H whose germ at the origin admits a Ck representative for each integer k, but no C∞ representative. A number theoretic condition on the coefficients of the Taylor series of H then insures the quasianalyticity of some differential algebras 𝒜n(H) induced by H. The o-minimality of the structure generated by H is deduced from this quasianalyticity property.

DOI : 10.5802/aif.2439
Classification : 03C64 57-99 26A27 57R45
Keywords: o-minimal, smooth cell decomposition
Mot clés : o-minimal, decomposition cellulaire lisse
Le Gal, Olivier 1 ; Rolin, Jean-Philippe 2

1 University of Toronto Department of Mathematics Toronto, Ontario M5S 2E4 (Canada)
2 Université de Bourgogne IMB, UFR Sciences et Techniques 9, Avenue Alain Savary BP 47870 21078 Dijon (France) 
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Le Gal, Olivier; Rolin, Jean-Philippe. An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition. Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 543-562. doi : 10.5802/aif.2439. https://www.numdam.org/articles/10.5802/aif.2439/

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